Asymptotic number of isometric generalized Fibonacci cubes

For a binary word f , let Qd(f) be the subgraph of the d-dimensional cube Qd induced on the set of all words that do not contain f as a factor. Let Gn be the set of words f of length n that are good in the sense that Qd(f) is isometric in Qd for all d. It is proved that limn→∞ |Gn|/2 exists. Estimates show that the limit is close to 0.08, that is, about eight percent of all words are good.

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Generalized Fibonacci cubes

Generalized Fibonacci cube Qd(f) is introduced as the graph obtained from the d-cube Qd by removing all vertices that contain a given binary string f as a substring. In this notation the Fibonacci cube Γd is Qd(11). The question whether Qd(f) is an isometric subgraph of Qd is studied. Embeddable and nonembeddable infinite series are given. The question is completely solved for strings f of leng...

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Asymptotic number of isometric generalized Fibonacci cubes

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